Classification of compact Lagrangian self-similar submanifolds with Legendrian capillary boundary in the unit ball

We classify smooth compact connected Lagrangian immersions $X$ in the closed unit ball of $\C^n$, $n\ge2$, satisfying $H+\varepsilon X^\perp=0$, $\varepsilon\in\{-1,0,1\}$, with Legendrian boundary on the unit sphere and constant contact angle on each connected component. We prove that the boundary has at most two connected components. When the boundary is connected, $X$ is a diffeomorphism onto an equatorial Lagrangian $n$-disk. When the boundary has two components, $X$ splits globally as $X(s,p)=γ(s)ψ(p)$, where $ψ$ is a compact minimal Legendrian immersion in the unit sphere and $γ$ is an Anciaux profile with a unique radial minimum. The two contact angles are supplementary. In complex dimension two, every non-disk solution is a finite cover of a Lagrangian catenoid segment for $\varepsilon=0$ or of a rotational Anciaux annulus for $\varepsilon=\pm1$. In higher complex dimensions, iterated Calabi suspensions produce families whose minimal Legendrian links have nontrivial topology.